Case study (4 Marks)

Question 14 Marks

Decimal form of rational numbers can be classified into two types.

Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form $\frac{\text{p}}{\text{q}}$, where p and q are co-prime and the prime factorisation of q is of the form $2^\text{n}\times5^\text{m},$ where n, rn are non-negative integers and vice-versa.
Let $\text{x}=\frac{\text{p}}{\text{q}}$ be a rational number, such that the prime factorisation of q is not of the form $2^\text{n}\times5^\text{m},$ where n and m are non-negative integers. Then x has a non-terminating repeating decimal expansion.

$\frac{441}{(2^{2}\times5^7\times7^2)}$ is which decimal?
$\frac{251}{(2^5\times\text{5}^3)}$ is which decimal?
does $\frac{15}{1600}$ have a terminating decimal expansion?
Or
$\frac{23}{(2^{5}\times5^3)}=$

Answer

1. $\frac{441}{(2^2\times5^7\times7^2)}=\frac{9}{(2^2\times5^7)}$ which is a terminating decimal.

2. Here denominator has only two prime factors i.e., 2 and 5 and hence it is a tenninating decimal.

3. Yes, $\frac{15}{1600}$ have a terminating decimal expansion
Or
$\frac{23}{(2^3\times5^2)}$
$=\frac{23}{200}=0.115$

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Question 24 Marks

HCF and LCM are widely used in number system especially in real numbers in finding relationship between different numbers and their general forms. Also, product of two positive integers is equal to the product of their HCF and LCM.

Based on the above information answer the following questions.

A boy with collection of marbles realizes that if he makes a group of 5 or 6 marbles, there are always two marbles left, then p will be odd, even, prime or not prime?
Find the least positive integer which on adding 1 is exactly divisible by 126 and 600.
Find the largest possible positive integer that will divide 398, 436 and 542 leaving remainder 7, 11, 15 respectively.
Or
If A, B and Care three rational numbers such that 85C - 340A = 109, 425A + 85B = 146, then the sum of A, B and C is divisible by?

Answer

1.Here, required numbers
= HCF (398 - 7,436 - 11,542 - 15)
= HCF (391, 425, 527) = 17

2.LCM of 126 and 600 = 2 × 3 × 21 × 100 = 12600.
The least positive integer which on adding 1 is exactly divisible by 126 and 600 = 12600 - 1 = 12599

3.Here, required numbers
= HCF (398 - 7,436 - 11,542 - 15)
= HCF (391, 425, 527) = 17
Or
Here 85C - 340A = 109 and 425A + 85B = 146 On adding them, we get,
85A + 85B + 85C = 255
⇒ A + 8 + C = 3, which is divisible by 3.

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Question 34 Marks

ln a classroom activity on real numbers, the students have to pick a number card from a pile and frame question on it if it is not a rational number for the rest of the class. The number cards picked up by first 5 students and their questions on the numbers for the rest of the class are as shown below. Answer them.

Ananya picked up $\sqrt{15}-\sqrt{10}$ and her question was $\sqrt{15}-\sqrt{10}$ is ______ number.
Suraj picked up $\sqrt{8}$ and his question was which type of number it was?
Preethi picked up $\sqrt{6}$ and her question was what will be irrational number of $\sqrt{6}$?
Or
Preethi picked up $\sqrt{9}$ and her question was what will be irrational number of $\sqrt{9}$?

Answer

1.Here $\sqrt{15}$ and $\sqrt{10}$ are both irrational and difference of two irrational numbers is also irrational.

2.Here $\sqrt{8}=2\sqrt{2}$ = product of rational and irrational numbers = irrational number.

3.$\sqrt{6}$ = 2.449
Or
$\sqrt{9}$ = 3

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Question 44 Marks

Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.
1. The statement 'One of every three consecutive positive integers is divisible by 3 ' is?
2. For what value of $n, 4^n$ ends in 0 ?
3. If $a$ is a positive rational number and $n$ is a positive integer greater than $I$, then for what value of $n, 4^n$ is a rational number?
Or
If n is any odd integer, then $n ^2-1$ is divisible by?

Answer

1. Let three consecutive positive integers be $n , n +1$ and $n +2$.
We know that when a number is divided by 3, the remainder obtained is either 0 or 1 or 2 .
So, $n=3 p$ or $3 p+1$ or $3 p+2$, where pis some integer.
If $n=3 p$, then 2 is divisible by 3 .
If $n=3 p+1$, then $n+2=3 p+1+2=3 p+3=3(p+1)$ is divisible by 3 .
If $n=3 p+2$, then $n+1=3 p+2+1=3 p+3=3(p+1)$ is divisible by 3 .
So, we can say that one of the numbers among $n, n+1$ and $n+2$ is always divisible by 3.2 . For a number to end in zero it must be divisible by 5 , but $4^n=2^{2 n}$ is never divisible by 5 . So, $4^n$ never ends in zero for any value of $n$.
3. We know that product of two rational numbers is also a rational number.
So, $a^2=a \times a=$ rational number.
$a^3=a^2 \times a=$ rational number.
$a^4=a^3 \times a=$ rational number.
$a^n=a^{n-1} \times a=$ rational number.
or
Any odd number is of the form of $(2 k+1)$, where $k$ is any integer.
So, $n ^2-1=(2 k +1)^2-1=4 k ^2+4 k$
For $k=1,4 k^2+4 k=8$, which is divisible by 8 .
Similarly, for $k=2,4 k^2+4 k=24$, which is divisible by 8 .
And fork $=3,4 k^2+4 k=48$, which is also divisible by 8 .
So, $4 k^2+4 k$ is divisible by 8 for all integers $k$, i.e., $n^2-1$ is divisible by 8 for all odd values of $n$.

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Question 54 Marks

Real numbers are extremely useful in everyday life. That is probably one of the main reasons we all learn how to count and add and subtract from a very young age. Real numbers help us to count and to measure out quantities of different items in various fields like retail, buying, catering, publishing etc. Every normal person uses real numbers in his daily life. After knowing the importance of real numbers, try and improve your knowledge about them by answering the following questions on real life based situations.

Two tankers contain 768 litres and 420 litres of fuel respectively. Find the maximum capacity of the container which can measure the fuel of either tanker exactly.
Pens are sold in pack of 8 and notepads are sold in pack of 12. Find the least number of pack of each type that one should buy so that there are equal number of pens and notepads.
Three people go for a morning walk together from the same place. Their steps measure 80cm, 85cm and 90cm respectively. What is the minimum distance travelled when they meet at first time after starting the walk assuming that their walking speed is same?
Or
ln a school Independence Day parade, a group of 594 students need to march behind a band of 189 members. The two groups have to march in the same number of columns. What is the maximum number of columns in which they can march?

Answer

1. Here $768=28 \times 3$ and $420=22 \times 3 \times 5 \times 7$
HCF of 768 and $420=22 \times 3=12$
So, the container which can measure fuel of either tanker exactly must be of 12 litres.2.LCM of 8 and 12 is 24 .
$\therefore$ The least number of pack of pens $=\frac{24}{8}=3$
$\therefore$ The least number of pack of note pads $=\frac{24}{12}=2$
3. Here $80=24 \times 5,85=17 \times 5$
and $90=2 \times 32 \times 5$
L.C.M of 80,85 and $90=24 \times 3 \times 3 \times 5 \times 17=12240$
Hence, the minimum distance each should walk when they at first time is 12240 cm .
Or
Here $594=2 \times 33 \times 11$ and $189=33 \times 7$
HCF of 594 and $189=3^3=27$
Hence, the maximum number of columns in which they can march is 27 .

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Maths Case study (4 Marks)

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